my dad is also a caucasian jewmountain jew and he wont identify as azeri they left in the s but he told me that they were always considered just jews as their nationality by other people and themselves in the caucasus they lived close together even here in new york they were able to form a community marriages usually also stayed within the community however overtime some married other groups but yes its mostly because of the religion and how others even viewed them when you ask them what they are they will say caucasian jews and come from azerbaijan